Laplace sequences of surfaces in projective space and two-dimensional Todaequations

We find that the Laplace sequences of surfaces of period n in projective sp ace Pn-1 have two types, while type II occurs only for even n. The integrab ility condition of the fundamental equations of these two types have the sa me form partial derivative (2)omega (i)/partial derivativex partial derivativet = - alpha (i-1)e(omegai-1) + 2 alpha (i)e(omegai) - alpha (i+1)e(omegai+1), alp ha (i) = +/-1 (i = 1, 2, ..., n). When all alpha (i) = 1, the above equations become two-dimensional Toda equ ations. Darboux transformations are used to obtain explicit solutions to th e above equations and the Laplace sequences of surfaces. Two examples in P- 3 of types I and II are constructed.